module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Homology.ModelCategory.Injective | {
"line": 157,
"column": 8
} | {
"line": 157,
"column": 56
} | {
"line": 158,
"column": 4
} | [
{
"pp": "case inr\nC : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Abelian C\nA : CochainComplex C ℤ\nhA : CochainComplex.plus C A\nB : CochainComplex C ℤ\nhB : CochainComplex.plus C B\nX : CochainComplex C ℤ\nhX : CochainComplex.plus C X\nY : CochainComplex C ℤ\nhY : CochainComplex.plus C Y\ni : A ⟶ B\n... | [] | exact this.acyclic_X₁ (by dsimp; infer_instance) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 181,
"column": 10
} | {
"line": 181,
"column": 12
} | {
"line": 181,
"column": 13
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝ : EnoughInjectives C\nn₀✝ n₁ : ℤ\nhn₁ : n₀✝ + 1 = n₁\nhf : ∀ i ≤ n₀✝, QuasiIsoAt f i\nn₀ : ℤ := n₁ - 1\nA : C\n⊢ ∀ (x₂ : A ⟶ K.X n₁),\n x₂ ≫ K.d n₁ (n₁ + 1) = 0 →\n ∀ (y₁ : A ⟶ (mid K L ... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝ : EnoughInjectives C\nn₀✝ n₁ : ℤ\nhn₁ : n₀✝ + 1 = n₁\nhf : ∀ i ≤ n₀✝, QuasiIsoAt f i\nn₀ : ℤ := n₁ - 1\nA : C\nx₁ : A ⟶ K.X n₁\n⊢ x₁ ≫ K.d n₁ (n₁ + 1) = 0 →\n ∀ (y₁ : A ⟶ (mid K L n₁).X n₀),\n x₁ ≫ ... | x₁ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Localization.Resolution | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 54
} | {
"line": 338,
"column": 2
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} D₁\ninst✝³ : Category.{v_4, u_4} D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₁' : MorphismProperty D₁\nW₂' : MorphismProperty D₂\nT : L... | [
"C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\ninst✝⁴ : Category.{v_3, u_3} D₁\ninst✝³ : Category.{v_4, u_4} D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₁' : MorphismProperty D₁\nW₂' : MorphismProperty D₂\nT : LocalizerMorp... | let ρ : T.LeftResolution X₂ := Classical.arbitrary _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.GuitartExact.Quotient | {
"line": 78,
"column": 6
} | {
"line": 79,
"column": 43
} | {
"line": 79,
"column": 43
} | [
{
"pp": "C₀ : Type u_1\nC : Type u_2\nH₀ : Type u_3\nH✝ : Type u_4\ninst✝⁶ : Category.{v_1, u_1} C₀\ninst✝⁵ : Category.{v_2, u_2} C\ninst✝⁴ : Category.{v_3, u_3} H₀\ninst✝³ : Category.{v_4, u_4} H✝\nT : C₀ ⥤ H₀\nL : C₀ ⥤ C\nR : H₀ ⥤ H✝\nB : C ⥤ H✝\ninst✝² : T.EssSurj\ninst✝¹ : T.Full\ninst✝ : B.Full\ne : T ⋙ R ... | [] | simp [R.map_comp, ← B.map_comp, dsimp% h.h₀, s₀.property,
dsimp% e.hom.naturality_assoc P.i₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.GuitartExact.Quotient | {
"line": 78,
"column": 6
} | {
"line": 79,
"column": 43
} | {
"line": 79,
"column": 43
} | [
{
"pp": "C₀ : Type u_1\nC : Type u_2\nH₀ : Type u_3\nH✝ : Type u_4\ninst✝⁶ : Category.{v_1, u_1} C₀\ninst✝⁵ : Category.{v_2, u_2} C\ninst✝⁴ : Category.{v_3, u_3} H₀\ninst✝³ : Category.{v_4, u_4} H✝\nT : C₀ ⥤ H₀\nL : C₀ ⥤ C\nR : H₀ ⥤ H✝\nB : C ⥤ H✝\ninst✝² : T.EssSurj\ninst✝¹ : T.Full\ninst✝ : B.Full\ne : T ⋙ R ... | [] | simp [R.map_comp, ← B.map_comp, dsimp% h.h₀, s₀.property,
dsimp% e.hom.naturality_assoc P.i₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GuitartExact.Quotient | {
"line": 78,
"column": 6
} | {
"line": 79,
"column": 43
} | {
"line": 79,
"column": 43
} | [
{
"pp": "C₀ : Type u_1\nC : Type u_2\nH₀ : Type u_3\nH✝ : Type u_4\ninst✝⁶ : Category.{v_1, u_1} C₀\ninst✝⁵ : Category.{v_2, u_2} C\ninst✝⁴ : Category.{v_3, u_3} H₀\ninst✝³ : Category.{v_4, u_4} H✝\nT : C₀ ⥤ H₀\nL : C₀ ⥤ C\nR : H₀ ⥤ H✝\nB : C ⥤ H✝\ninst✝² : T.EssSurj\ninst✝¹ : T.Full\ninst✝ : B.Full\ne : T ⋙ R ... | [] | simp [R.map_comp, ← B.map_comp, dsimp% h.h₀, s₀.property,
dsimp% e.hom.naturality_assoc P.i₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived | {
"line": 126,
"column": 6
} | {
"line": 127,
"column": 75
} | {
"line": 128,
"column": 4
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nH : Type u₃\ninst✝¹⁰ : Category.{v₁, u₁} C₁\ninst✝⁹ : Category.{v₂, u₂} C₂\ninst✝⁸ : Category.{v₃, u₃} H\nD₁ : Type u₄\nD₂ : Type u₅\ninst✝⁷ : Category.{v₄, u₄} D₁\ninst✝⁶ : Category.{v₅, u₅} D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂... | [
"C₁ : Type u₁\nC₂ : Type u₂\nH : Type u₃\ninst✝¹⁰ : Category.{v₁, u₁} C₁\ninst✝⁹ : Category.{v₂, u₂} C₂\ninst✝⁸ : Category.{v₃, u₃} H\nD₁ : Type u₄\nD₂ : Type u₅\ninst✝⁷ : Category.{v₄, u₄} D₁\ninst✝⁶ : Category.{v₅, u₅} D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\nL₁ : C₁ ⥤ ... | ← isIso_comp_right_iff (α₁.app X)
((Φ.rightDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂).app (L₁.obj X)), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Preadditive.Injective.InjectiveObject | {
"line": 35,
"column": 4
} | {
"line": 39,
"column": 63
} | {
"line": 41,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : Type u_1\n⊢ (isInjective C).limitsOfShape (Discrete J) ≤ isInjective C",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Injective",
"C... | [] | rintro Y ⟨p⟩
have (j : J) : Injective (p.diag.obj ⟨j⟩) := p.prop_diag_obj _
exact ⟨fun q i _ ↦ ⟨p.isLimit.lift (Cone.mk _
(Discrete.natTrans (fun ⟨j⟩ ↦ (Injective.factorThru (q ≫ p.π.app ⟨j⟩) i :)))),
p.isLimit.hom_ext (fun ⟨j⟩ ↦ by simp [p.isLimit.fac])⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Preadditive.Injective.InjectiveObject | {
"line": 35,
"column": 4
} | {
"line": 39,
"column": 63
} | {
"line": 41,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : Type u_1\n⊢ (isInjective C).limitsOfShape (Discrete J) ≤ isInjective C",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Injective",
"C... | [] | rintro Y ⟨p⟩
have (j : J) : Injective (p.diag.obj ⟨j⟩) := p.prop_diag_obj _
exact ⟨fun q i _ ↦ ⟨p.isLimit.lift (Cone.mk _
(Discrete.natTrans (fun ⟨j⟩ ↦ (Injective.factorThru (q ≫ p.π.app ⟨j⟩) i :)))),
p.isLimit.hom_ext (fun ⟨j⟩ ↦ by simp [p.isLimit.fac])⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.ExactFunctor | {
"line": 122,
"column": 57
} | {
"line": 125,
"column": 16
} | {
"line": 127,
"column": 0
} | [
{
"pp": "C₁ : Type u₁\ninst✝¹² : Category.{v₁, u₁} C₁\ninst✝¹¹ : Abelian C₁\ninst✝¹⁰ : HasDerivedCategory C₁\nC₂ : Type u₂\ninst✝⁹ : Category.{v₂, u₂} C₂\ninst✝⁸ : Abelian C₂\ninst✝⁷ : HasDerivedCategory C₂\nF : C₁ ⥤ C₂\ninst✝⁶ : F.Additive\ninst✝⁵ : PreservesFiniteLimits F\ninst✝⁴ : PreservesFiniteColimits F\n... | [] | by
rw [← Localization.functor_linear_iff DerivedCategory.Qh (HomotopyCategory.quasiIso C₁
(ComplexShape.up ℤ)) R ((F.mapHomotopyCategory (ComplexShape.up ℤ)).comp DerivedCategory.Qh)]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.DerivedCategory.DerivabilityStructureInjectives | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 93
} | {
"line": 206,
"column": 93
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nK L : CochainComplex.Plus (InjectiveObject C)\nf : K ⟶ L\n⊢ IsIso ((ι (InjectiveObject C)).map ((quotient (InjectiveObject C)).map f)) ↔\n CochainComplex.Plus.quasiIso C ((InjectiveObject.ι C).mapCochainComplexPlus.map f)",
"ppTerm... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nK L : CochainComplex.Plus (InjectiveObject C)\nf : K ⟶ L\n⊢ IsIso\n (((InjectiveObject.ι C).mapHomotopyCategory (ComplexShape.up ℤ)).map\n ((ι (InjectiveObject C)).map ((quotient (InjectiveObject C)).map f))) ↔\n CochainComplex.Plus.qu... | ← isIso_iff_of_reflects_iso _ (Functor.mapHomotopyCategory (InjectiveObject.ι C) (.up ℤ)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.DerivedCategory.DerivabilityStructureInjectives | {
"line": 220,
"column": 2
} | {
"line": 223,
"column": 86
} | {
"line": 225,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nK L : CochainComplex.Plus (InjectiveObject C)\nf : K ⟶ L\n⊢ (CochainComplex.Plus.quasiIso C).inverseImage (InjectiveObject.ι C).mapCochainComplexPlus f ↔\n (homotopyEquivalences (InjectiveObject C) (ComplexShape.up ℤ)).inverseImage\n ... | [] | simp [CochainComplex.Plus.quasiIso, Functor.mapCochainComplexPlus,
← HomologicalComplex.isIso_quotient_map_iff_homotopyEquivalences,
CochainComplex.IsKInjective.quasiIso_iff,
← isIso_iff_of_reflects_iso _ ((InjectiveObject.ι C).mapHomotopyCategory (.up ℤ))] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.Functor | {
"line": 54,
"column": 8
} | {
"line": 54,
"column": 20
} | {
"line": 54,
"column": 21
} | [
{
"pp": "T : Type u_1\ninst✝² : Category.{v_1, u_1} T\nV : Type u_2\ninst✝¹ : Category.{v_2, u_2} V\ninst✝ : HasZeroMorphisms V\nι : Type u_3\nc : ComplexShape ι\nC : HomologicalComplex (T ⥤ V) c\nt : T\ni j : ι\nh : ¬c.Rel i j\nthis : ∀ (x : T), (C.d i j).app x = NatTrans.app 0 x\n⊢ (C.d i j).app t = 0",
"... | [] | exact this t | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.Functor | {
"line": 50,
"column": 8
} | {
"line": 50,
"column": 20
} | {
"line": 51,
"column": 6
} | [
{
"pp": "T : Type u_1\ninst✝² : Category.{v_1, u_1} T\nV : Type u_2\ninst✝¹ : Category.{v_2, u_2} V\ninst✝ : HasZeroMorphisms V\nι : Type u_3\nc : ComplexShape ι\nC : HomologicalComplex (T ⥤ V) c\nt : T\ni j k : ι\nx✝¹ : c.Rel i j\nx✝ : c.Rel j k\nthis : ∀ (x : T), (C.d i j ≫ C.d j k).app x = NatTrans.app 0 x\n... | [] | exact this t | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 313,
"column": 45
} | {
"line": 313,
"column": 82
} | {
"line": 313,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q : ℤ\nf : K.X p ⟶ X\nn : ℤ\nh : p + n = q\np' : ℤ\nhp' : p' + 1 = p\nhf : K.d p' p ≫ f = 0\nX' : C\ng : X ⟶ X'\n⊢ (Cochain.toSingleEquiv h) ↑(toSingleMk (f ≫ g) h p' hp' ⋯) =\n ... | [] | by simp [Cochain.toSingleMk_postcomp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Idempotents.FunctorExtension | {
"line": 240,
"column": 81
} | {
"line": 248,
"column": 72
} | {
"line": 250,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Category.{v_3, u_3} E\ninst✝ : IsIdempotentComplete D\n⊢ ((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).IsEquivalence",
"ppTerm": "?m.15",
"assigned": true,
"usedConstant... | [] | by
have : ((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C) ⋙
(whiskeringRight C D (Karoubi D)).obj (toKaroubi D) ⋙
(whiskeringRight C (Karoubi D) D).obj (Functor.inv (toKaroubi D))).IsEquivalence := by
change (karoubiUniversal C D).inverse.IsEquivalence
infer_instance
exact Functor.isEquivalence_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.Basic | {
"line": 211,
"column": 77
} | {
"line": 214,
"column": 28
} | {
"line": 216,
"column": 0
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Category.{u_4, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\ni₀ i₁ i₂ : ι\nf : i₀ ⟶ i₁\ng : i₁ ⟶ i₂\nfg : i₀ ⟶ i₂\nhfg : f ≫ g = fg\nh₁ : IsZero ((X.H n₀).obj (mk₁ f))\nh₂ : IsZero ((X.H n₁).obj (mk₁... | [] | by
have := X.mono_H_map_twoδ₁Toδ₀ n₀ f g fg hfg h₁
have := X.epi_H_map_twoδ₁Toδ₀ n₀ n₁ hn₁ f g fg hfg h₂
apply isIso_of_mono_of_epi | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.Differentials | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 11
} | {
"line": 131,
"column": 2
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nf₂₃ : j ⟶ l\nh₂₃ : f₂ ≫ f₃ = f₂₃\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ X.toCycles f₂ f₃ f₂₃ h₂₃ n₀ ≫ X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁ ... | [
"C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni j k l : ι\nf₁ : i ⟶ j\nf₂ : j ⟶ k\nf₃ : k ⟶ l\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\n⊢ X.toCycles f₂ f₃ (f₂ ≫ f₃) ⋯ n₀ ≫ X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁ = X.δ f₁ (f₂ ≫ f₃) n₀ n₁ hn₁ ≫ X.pOpcycles... | subst h₂₃ | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Order.WithBotTop | {
"line": 36,
"column": 61
} | {
"line": 36,
"column": 73
} | {
"line": 37,
"column": 0
} | [
{
"pp": "ι : Type u_1\na : ι\n⊢ coe a ≠ ⊥",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"WithBotTop.coe",
"False",
"Option.ctorIdx",
"HEq.refl",
"False.elim",
"noConfusion_of_Nat",
"Option.some",
"Eq.casesOn",
"Bot.bot",
"WithTop... | [] | by rintro ⟨⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.WithBotTop | {
"line": 37,
"column": 61
} | {
"line": 37,
"column": 73
} | {
"line": 38,
"column": 0
} | [
{
"pp": "ι : Type u_1\na : ι\n⊢ coe a ≠ ⊤",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"WithBotTop.coe",
"False",
"Option.some.noConfusion",
"Option.ctorIdx",
"HEq.refl",
"False.elim",
"noConfusion_of_Nat",
"Option.some",
"Eq.casesOn"... | [] | by rintro ⟨⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.WithBotTop | {
"line": 38,
"column": 53
} | {
"line": 38,
"column": 65
} | {
"line": 40,
"column": 0
} | [
{
"pp": "ι : Type u_1\n⊢ ⊤ ≠ ⊥",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"False",
"Option.ctorIdx",
"HEq.refl",
"False.elim",
"noConfusion_of_Nat",
"Option.some",
"Eq.casesOn",
"WithBot.instTop",
"Bot.bot",
"Option.none",
... | [] | by rintro ⟨⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.Differentials | {
"line": 238,
"column": 63
} | {
"line": 243,
"column": 44
} | {
"line": 245,
"column": 0
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni₀ i₁ i₂ : ι\nf₁ : i₀ ⟶ i₁\nf₂ : i₁ ⟶ i₂\nn₀ n₁ n₂ n₃ : ℤ\nhn₁ : n₀ + 1 = n₁\nhn₂ : n₁ + 1 = n₂\nhn₃ : n₂ + 1 = n₃\n⊢ X.d (𝟙 i₀) f₁ (𝟙 i₁) f₂ (𝟙 i₂) n₀ n₁ n₂ n₃ hn₁ ... | [] | by
rw [← cancel_epi (X.πE (𝟙 i₁) f₂ (𝟙 i₂) n₀ n₁ n₂ hn₁ hn₂),
← cancel_epi (X.toCycles (𝟙 i₁) f₂ f₂ (by simp) n₁),
X.toCycles_πE_d_assoc (𝟙 i₀) f₁ (𝟙 i₁) f₂ (𝟙 i₂) f₁ (by simp) _ _ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃,
πE_EIsoH_hom .., πE_EIsoH_hom_assoc .., cyclesIsoH_inv_hom_id ..,
comp_id, cyclesIsoH_inv_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 294,
"column": 4
} | {
"line": 300,
"column": 38
} | {
"line": 301,
"column": 4
} | [
{
"pp": "case pos\nC : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq' pq'' : κ\nhpq' : (c r).next pq' = pq... | [
"case pos\nC : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq' pq'' : κ\nhpq' : (c r).next pq' = pq''\ni₀' i₀ i... | refine ShortComplex.exact_of_iso (Iso.symm ?_)
(X.dKernelSequence_exact
(homOfLE (show data.i₀ r pq'' ≤ i₀' by
simpa only [hi₀', data.i₀_prev r r' _ _ h] using data.le₀₁ r pq''))
(homOfLE (data.i₀_le' hrr' hr pq' hi₀' hi₀)) (homOfLE (data.le₀₁' r hr pq' hi₀ hi₁))
(homOfLE (data.l... | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 379,
"column": 2
} | {
"line": 379,
"column": 11
} | {
"line": 380,
"column": 2
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).prev pq' = pq\ni₀ i₁ i₂ i₃... | [
"C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq' : κ\ni₀ i₁ i₂ i₃ i₃' : ι\nhi₀ : i₀ = data.i₀ r pq' ⋯\nhi₁ ... | subst hpq | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 376,
"column": 2
} | {
"line": 380,
"column": 12
} | {
"line": 382,
"column": 0
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).prev pq' = pq\ni₀ i₁ i₂ i₃... | [] | apply X.isIso_map_fourδ₄Toδ₃_of_isZero _ _ _ _ _ _ _ _ _ _
refine X.isZero_H_obj_mk₁_i₃_le' data r r' hrr' hr pq' (fun _ hk ↦ ?_) _ (by lia) _ _ hi₃ hi₃'
obtain rfl := (c r).prev_eq' hk
subst hpq
exact h hk | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 376,
"column": 2
} | {
"line": 380,
"column": 12
} | {
"line": 382,
"column": 0
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).prev pq' = pq\ni₀ i₁ i₂ i₃... | [] | apply X.isIso_map_fourδ₄Toδ₃_of_isZero _ _ _ _ _ _ _ _ _ _
refine X.isZero_H_obj_mk₁_i₃_le' data r r' hrr' hr pq' (fun _ hk ↦ ?_) _ (by lia) _ _ hi₃ hi₃'
obtain rfl := (c r).prev_eq' hk
subst hpq
exact h hk | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 516,
"column": 16
} | {
"line": 516,
"column": 49
} | {
"line": 516,
"column": 49
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).Rel pq pq'\n... | [] | rw [h₂, ← data.hc₀₂ r pq pq' hpq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 516,
"column": 16
} | {
"line": 516,
"column": 49
} | {
"line": 516,
"column": 49
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).Rel pq pq'\n... | [] | rw [h₂, ← data.hc₀₂ r pq pq' hpq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 516,
"column": 16
} | {
"line": 516,
"column": 49
} | {
"line": 516,
"column": 49
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).Rel pq pq'\n... | [] | rw [h₂, ← data.hc₀₂ r pq pq' hpq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Submodule | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 41
} | {
"line": 226,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\np : Submodule R M\n⊢ (∀ (x : L), ∀ m ∈ p, ⁅x, m⁆ ∈ p) → ∃ N, ↑N = p",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
... | [] | intro h; use { p with lie_mem := @h } | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Submodule | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 41
} | {
"line": 226,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\np : Submodule R M\n⊢ (∀ (x : L), ∀ m ∈ p, ⁅x, m⁆ ∈ p) → ∃ N, ↑N = p",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
... | [] | intro h; use { p with lie_mem := @h } | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Submodule | {
"line": 283,
"column": 64
} | {
"line": 284,
"column": 41
} | {
"line": 286,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN : LieSubmodule R L M\n⊢ ↑N = ⊥ ↔ N = ⊥",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"... | [] | by
rw [← toSubmodule_inj, bot_toSubmodule] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Submodule | {
"line": 287,
"column": 49
} | {
"line": 288,
"column": 41
} | {
"line": 290,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN : Submodule R M\nh : ∀ {x : L} {m : M}, m ∈ N.carrier → ⁅x, m⁆ ∈ N.carrier\n⊢ { toSubmodule := N, lie_mem := h } = ⊥ ↔ N = ⊥",
"ppTerm": "?m.39",
... | [] | by
rw [← toSubmodule_inj, bot_toSubmodule] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 91,
"column": 34
} | {
"line": 91,
"column": 38
} | {
"line": 91,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Submodule R M\nx₁ x₂ y₁ y₂ : M\nhxy₁ : Submodule.Quotient.mk x₁ = Submodule.Quotient.mk y₁\nhxy₂ : Submodule.Quotient.mk x₂ = Submodule.Quotient.mk y₂\n⊢ Submodule.Quotient.mk y₁ + Submodule.Quotient.mk x₂ = S... | [] | hxy₂ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 121,
"column": 12
} | {
"line": 121,
"column": 16
} | {
"line": 121,
"column": 16
} | [
{
"pp": "A : Type u_3\ninst✝ : CommRing A\nI : Ideal A\nx₁ x₂ y₁ y₂ : A\nhxy₁ : (Ideal.Quotient.mk I) x₁ = (Ideal.Quotient.mk I) y₁\nhxy₂ : (Ideal.Quotient.mk I) x₂ = (Ideal.Quotient.mk I) y₂\n⊢ (Ideal.Quotient.mk I) y₁ * (Ideal.Quotient.mk I) x₂ = (Ideal.Quotient.mk I) y₁ * (Ideal.Quotient.mk I) y₂",
"ppTe... | [
"A : Type u_3\ninst✝ : CommRing A\nI : Ideal A\nx₁ x₂ y₁ y₂ : A\nhxy₁ : (Ideal.Quotient.mk I) x₁ = (Ideal.Quotient.mk I) y₁\nhxy₂ : (Ideal.Quotient.mk I) x₂ = (Ideal.Quotient.mk I) y₂\n⊢ (Ideal.Quotient.mk I) y₁ * (Ideal.Quotient.mk I) y₂ = (Ideal.Quotient.mk I) y₁ * (Ideal.Quotient.mk I) y₂"
] | hxy₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 145,
"column": 34
} | {
"line": 145,
"column": 38
} | {
"line": 145,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Submodule R M\nx₁ x₂ y₁ y₂ : M\nhxy₁ : Submodule.Quotient.mk x₁ = Submodule.Quotient.mk y₁\nhxy₂ : Submodule.Quotient.mk x₂ = Submodule.Quotient.mk y₂\n⊢ Submodule.Quotient.mk y₁ - Submodule.Quotient.mk x₂ = S... | [] | hxy₂ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 349,
"column": 65
} | {
"line": 349,
"column": 86
} | {
"line": 349,
"column": 86
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\ninst✝ : Algebra K L\nσ : L →ₐ[K] L\nhσ : IsOfFinOrder σ\n⊢ orderOf σ = orderOf ↑hσ.unit",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"congrArg",
... | [
"K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\ninst✝ : Algebra K L\nσ : L →ₐ[K] L\nhσ : IsOfFinOrder σ\n⊢ orderOf σ = orderOf σ",
"case hf\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\ninst✝ : Algebra K L\nσ : L →ₐ[K] L\nhσ : IsO... | IsOfFinOrder.val_unit | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 83,
"column": 10
} | {
"line": 83,
"column": 12
} | {
"line": 84,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R\n⊢ a ∈ I → ((Quotient.mk (map C I)).comp C) a = 0",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"CommSemiring.toSemiring",
"Membership.mem",
"Ideal",
"CommRing.toCommSemiring"... | [
"R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R\nha : a ∈ I\n⊢ ((Quotient.mk (map C I)).comp C) a = 0"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 89,
"column": 10
} | {
"line": 89,
"column": 12
} | {
"line": 90,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R[X]\n⊢ a ∈ map C I → (eval₂RingHom (C.comp (Quotient.mk I)) X) a = 0",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Semiring.toModule",
"CommSemiring.toSemiring",
"RingHom",
"Member... | [
"R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R[X]\nha : a ∈ map C I\n⊢ (eval₂RingHom (C.comp (Quotient.mk I)) X) a = 0"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.PowerBasis | {
"line": 272,
"column": 97
} | {
"line": 273,
"column": 56
} | {
"line": 275,
"column": 0
} | [
{
"pp": "S : Type u_2\ninst✝⁴ : Ring S\nA : Type u_4\ninst✝³ : CommRing A\ninst✝² : Algebra A S\nS' : Type u_7\ninst✝¹ : Ring S'\ninst✝ : Algebra A S'\npb : PowerBasis A S\ny : S'\nhy : (aeval y) (minpoly A pb.gen) = 0\nx : A\n⊢ ((pb.basis.constr A) fun i ↦ y ^ ↑i) ((algebraMap A S) x) = (algebraMap A S') x",
... | [] | by
convert! pb.constr_pow_aeval hy (C x) <;> rw [aeval_C] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Squarefree.Basic | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 89
} | {
"line": 103,
"column": 90
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Monoid R\nx y : R\nn : ℕ\nhx : Squarefree y\nh : x ^ n ∣ y\nhu : ¬IsUnit x\n⊢ x ^ n ∣ x",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"semigroupDvd",
"Squarefree.eq_zero_or_one_of_pow_of_not_isUnit",
"instOfNatNa... | [
"case neg.inl\nR : Type u_1\ninst✝ : Monoid R\nx y : R\nhx : Squarefree y\nhu : ¬IsUnit x\nh : x ^ 0 ∣ y\n⊢ x ^ 0 ∣ x",
"case neg.inr\nR : Type u_1\ninst✝ : Monoid R\nx y : R\nhx : Squarefree y\nhu : ¬IsUnit x\nh : x ^ 1 ∣ y\n⊢ x ^ 1 ∣ x"
] | rcases (hx.squarefree_of_dvd h).eq_zero_or_one_of_pow_of_not_isUnit hu with rfl | rfl | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.Squarefree.Basic | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 39
} | {
"line": 179,
"column": 39
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : IsCancelMulZero R\nz w : R\nh0 : z * z * w ≠ 0\nh : z * (z * w) ∣ 1 * (z * w)\n⊢ z * w ≠ 0",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semigroup.toMul",
"Dvd.dvd",
"HMul.hMul",
... | [
"R : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : IsCancelMulZero R\nz w : R\nh0 : z ≠ 0 ∧ z * w ≠ 0\nh : z * (z * w) ∣ 1 * (z * w)\n⊢ z * w ≠ 0"
] | rw [mul_assoc, mul_ne_zero_iff] at h0 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Separable | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 27
} | {
"line": 72,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\na : R\n⊢ IsCoprime (X + C a) 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"CommSemiring.toSemiring",
"RingHom",
"Polynomial.instAdd",
"Polynomial",
"instHAdd",
"RingHom.instFunLike",... | [] | exact isCoprime_one_right | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Separable | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 27
} | {
"line": 76,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\n⊢ IsCoprime X 1",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"Polynomial",
"isCoprime_one_right",
"Polynomial.commSemiring",
"Polynomial.X"
],
"usedFVars": [
"R",
"i... | [] | exact isCoprime_one_right | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Separable | {
"line": 339,
"column": 2
} | {
"line": 352,
"column": 58
} | {
"line": 354,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\n⊢ f.Separable ∨ ¬f.Separable ∧ ∃ g, Irreducible g ∧ (expand F p) g = f",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Polynomial.derivative",
"WithBot.instPreorder",
... | [] | classical
exact if H : derivative f = 0 then by
rcases p.eq_zero_or_pos with (rfl | hp)
· have := CharP.charP_to_charZero F
have := derivative_eq_zero.1 H
have := (natDegree_pos_iff_degree_pos.mpr <| degree_pos_of_irreducible hf).ne'
contradiction
have := isLocalHom_expand F hp
exact... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.FieldTheory.Separable | {
"line": 339,
"column": 2
} | {
"line": 352,
"column": 58
} | {
"line": 354,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\n⊢ f.Separable ∨ ¬f.Separable ∧ ∃ g, Irreducible g ∧ (expand F p) g = f",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Polynomial.derivative",
"WithBot.instPreorder",
... | [] | classical
exact if H : derivative f = 0 then by
rcases p.eq_zero_or_pos with (rfl | hp)
· have := CharP.charP_to_charZero F
have := derivative_eq_zero.1 H
have := (natDegree_pos_iff_degree_pos.mpr <| degree_pos_of_irreducible hf).ne'
contradiction
have := isLocalHom_expand F hp
exact... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Separable | {
"line": 339,
"column": 2
} | {
"line": 352,
"column": 58
} | {
"line": 354,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\n⊢ f.Separable ∨ ¬f.Separable ∧ ∃ g, Irreducible g ∧ (expand F p) g = f",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Polynomial.derivative",
"WithBot.instPreorder",
... | [] | classical
exact if H : derivative f = 0 then by
rcases p.eq_zero_or_pos with (rfl | hp)
· have := CharP.charP_to_charZero F
have := derivative_eq_zero.1 H
have := (natDegree_pos_iff_degree_pos.mpr <| degree_pos_of_irreducible hf).ne'
contradiction
have := isLocalHom_expand F hp
exact... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Perfect | {
"line": 396,
"column": 2
} | {
"line": 400,
"column": 56
} | {
"line": 402,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\np n : ℕ\ninst✝¹ : ExpChar R p\nf : R[X]\ninst✝ : PerfectRing R p\n⊢ ((expand R (p ^ n)) f).roots = p ^ n • Multiset.map (⇑(iterateFrobeniusEquiv R p n).symm) f.roots",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"iter... | [] | classical
refine ext' fun r ↦ ?_
rw [count_roots, rootMultiplicity_expand_pow, ← count_roots, count_nsmul, count_map,
count_eq_card_filter_eq]; congr; ext
exact (iterateFrobeniusEquiv R p n).eq_symm_apply.symm | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.FieldTheory.Perfect | {
"line": 396,
"column": 2
} | {
"line": 400,
"column": 56
} | {
"line": 402,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\np n : ℕ\ninst✝¹ : ExpChar R p\nf : R[X]\ninst✝ : PerfectRing R p\n⊢ ((expand R (p ^ n)) f).roots = p ^ n • Multiset.map (⇑(iterateFrobeniusEquiv R p n).symm) f.roots",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"iter... | [] | classical
refine ext' fun r ↦ ?_
rw [count_roots, rootMultiplicity_expand_pow, ← count_roots, count_nsmul, count_map,
count_eq_card_filter_eq]; congr; ext
exact (iterateFrobeniusEquiv R p n).eq_symm_apply.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Perfect | {
"line": 396,
"column": 2
} | {
"line": 400,
"column": 56
} | {
"line": 402,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\np n : ℕ\ninst✝¹ : ExpChar R p\nf : R[X]\ninst✝ : PerfectRing R p\n⊢ ((expand R (p ^ n)) f).roots = p ^ n • Multiset.map (⇑(iterateFrobeniusEquiv R p n).symm) f.roots",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"iter... | [] | classical
refine ext' fun r ↦ ?_
rw [count_roots, rootMultiplicity_expand_pow, ← count_roots, count_nsmul, count_map,
count_eq_card_filter_eq]; congr; ext
exact (iterateFrobeniusEquiv R p n).eq_symm_apply.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdjoinRoot | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 34
} | {
"line": 184,
"column": 2
} | [
{
"pp": "R : Type u_1\nT : Type u_3\ninst✝¹ : CommRing R\ninst✝ : Semiring T\np : R[X]\nf g : AdjoinRoot p →+* T\nhAlg : f.comp (of p) = g.comp (of p)\nhRoot : f (root p) = g (root p)\n⊢ f = g",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.ringHom_ext",
"CommSe... | [
"R : Type u_1\nT : Type u_3\ninst✝¹ : CommRing R\ninst✝ : Semiring T\np : R[X]\nf g : AdjoinRoot p →+* T\nhAlg : f.comp (of p) = g.comp (of p)\nhRoot : f (root p) = g (root p)\n⊢ f.comp (Ideal.Quotient.mk (span {p})) = g.comp (Ideal.Quotient.mk (span {p}))"
] | apply Ideal.Quotient.ringHom_ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.AdjoinRoot | {
"line": 253,
"column": 14
} | {
"line": 253,
"column": 95
} | {
"line": 255,
"column": 0
} | [
{
"pp": "case ih\nR : Type u_1\ninst✝ : CommRing R\nf p : R[X]\n⊢ (mk f) p ∈ R[root f]",
"ppTerm": "?ih",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"AdjoinRoot",
"CommSemiring.toSemiring",
"Polynomial.algebraOfAlgebra",
"Algebra.adjoin",
"Ring... | [] | exact (Algebra.adjoin_singleton_eq_range_aeval R (root f)).symm ▸ ⟨p, aeval_eq p⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.AdjoinRoot | {
"line": 253,
"column": 14
} | {
"line": 253,
"column": 95
} | {
"line": 255,
"column": 0
} | [
{
"pp": "case ih\nR : Type u_1\ninst✝ : CommRing R\nf p : R[X]\n⊢ (mk f) p ∈ R[root f]",
"ppTerm": "?ih",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"AdjoinRoot",
"CommSemiring.toSemiring",
"Polynomial.algebraOfAlgebra",
"Algebra.adjoin",
"Ring... | [] | exact (Algebra.adjoin_singleton_eq_range_aeval R (root f)).symm ▸ ⟨p, aeval_eq p⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AdjoinRoot | {
"line": 253,
"column": 14
} | {
"line": 253,
"column": 95
} | {
"line": 255,
"column": 0
} | [
{
"pp": "case ih\nR : Type u_1\ninst✝ : CommRing R\nf p : R[X]\n⊢ (mk f) p ∈ R[root f]",
"ppTerm": "?ih",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"AdjoinRoot",
"CommSemiring.toSemiring",
"Polynomial.algebraOfAlgebra",
"Algebra.adjoin",
"Ring... | [] | exact (Algebra.adjoin_singleton_eq_range_aeval R (root f)).symm ▸ ⟨p, aeval_eq p⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.AdjointAction.Basic | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 95
} | {
"line": 46,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nh : IsNilpotent a\nhl : IsNilpotent (LinearMap.mulLeft R a)\n⊢ IsNilpotent ((LinearMap.mulLeft R - LinearMap.mulRight R) a)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"LieAlgeb... | [
"R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nh : IsNilpotent a\nhl : IsNilpotent (LinearMap.mulLeft R a)\nhr : IsNilpotent (LinearMap.mulRight R a)\n⊢ IsNilpotent ((LinearMap.mulLeft R - LinearMap.mulRight R) a)"
] | have hr : IsNilpotent (LinearMap.mulRight R a) := by rwa [LinearMap.isNilpotent_mulRight_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Semisimple | {
"line": 130,
"column": 25
} | {
"line": 130,
"column": 66
} | {
"line": 130,
"column": 66
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nhn : IsNilpotent f\nhs : f.IsSemisimple\nn : ℕ\nh0 : X ^ n ∈ RingHom.ker (aeval f)\n⊢ X ∈ RingHom.ker (aeval f)",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nhn : IsNilpotent f\nhs : f.IsSemisimple\nn : ℕ\nh0 : X ^ n ∈ annihilator R[X] (AEval R M f)\n⊢ X ∈ annihilator R[X] (AEval R M f)"
] | ← AEval.annihilator_eq_ker_aeval (M := M) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 88,
"column": 2
} | {
"line": 90,
"column": 33
} | {
"line": 91,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := max n₁ n₂\nh₄ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\n⊢ exp (a + b) = exp a * exp b",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"Eq... | [
"A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := max n₁ n₂\nh₄ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\n⊢ ∑ i ∈ range (2 * N + 1), (↑i !)⁻¹ • (a + b) ^ i =\n (∑ i ∈ range (N + 1), (↑i !)⁻¹ • a ^ i) * ∑ i ∈ range (N + 1), (... | rw [exp_eq_sum (k := 2 * N + 1)
(Commute.add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero h₁ h₄ h₅ (by lia)),
exp_eq_sum h₄, exp_eq_sum h₅] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.AdjoinRoot | {
"line": 957,
"column": 4
} | {
"line": 957,
"column": 36
} | {
"line": 957,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nf p : R[X]\n⊢ (quotMapOfEquivQuotMapCMapMk I f).symm\n ((quotMapCMapSpanMkEquivQuotMapCQuotMapMk I f).symm\n ((quotEquivOfEq ⋯).symm\n ((Polynomial.quotQuotEquivComm I f)\n ((Ideal.Quotient.mk (span {Polynomial.map (Ideal.Qu... | [
"R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nf p : R[X]\n⊢ (quotMapOfEquivQuotMapCMapMk I f).symm\n ((quotMapCMapSpanMkEquivQuotMapCQuotMapMk I f).symm\n ((quotEquivOfEq ⋯).symm\n ((Ideal.Quotient.mk (span {(Ideal.Quotient.mk (Ideal.map C I)) f}))\n ((Ideal.Quotient.mk (Ideal.map ... | Polynomial.quotQuotEquivComm_mk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 987,
"column": 2
} | {
"line": 988,
"column": 64
} | {
"line": 990,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nI : Ideal R\n⊢ (quotEquivQuotMap f I).symm\n ((Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) f})) (Polynomial.map (Ideal.Quotient.mk I) g)) =\n (Ideal.Quotient.mk (Ideal.map (of f) I)) ((mk f) g)",
"ppTerm": "?m.45",
"assigne... | [] | rw [AdjoinRoot.quotEquivQuotMap_symm_apply,
AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot_symm_mk_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.AdjoinRoot | {
"line": 987,
"column": 2
} | {
"line": 988,
"column": 64
} | {
"line": 990,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nI : Ideal R\n⊢ (quotEquivQuotMap f I).symm\n ((Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) f})) (Polynomial.map (Ideal.Quotient.mk I) g)) =\n (Ideal.Quotient.mk (Ideal.map (of f) I)) ((mk f) g)",
"ppTerm": "?m.45",
"assigne... | [] | rw [AdjoinRoot.quotEquivQuotMap_symm_apply,
AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot_symm_mk_mk] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AdjoinRoot | {
"line": 987,
"column": 2
} | {
"line": 988,
"column": 64
} | {
"line": 990,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nI : Ideal R\n⊢ (quotEquivQuotMap f I).symm\n ((Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) f})) (Polynomial.map (Ideal.Quotient.mk I) g)) =\n (Ideal.Quotient.mk (Ideal.map (of f) I)) ((mk f) g)",
"ppTerm": "?m.45",
"assigne... | [] | rw [AdjoinRoot.quotEquivQuotMap_symm_apply,
AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot_symm_mk_mk] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Divisibility.Lemmas | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 48
} | {
"line": 55,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\nx y : R\nn m p : ℕ\ninst✝ : Semiring R\nhp : n + m ≤ p + 1\nh_comm : Commute x y\nhy : y ^ n = 0\nx✝ : ℕ × ℕ\ni j : ℕ\nhij : i + j = p\nhi : i + 1 ≤ m\n⊢ x ^ m ∣ x ^ (i, j).1 * y ^ (i, j).2",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
... | [] | · simp [pow_eq_zero_of_le (by lia : n ≤ j) hy] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.BaseChange | {
"line": 99,
"column": 12
} | {
"line": 99,
"column": 35
} | {
"line": 99,
"column": 36
} | [
{
"pp": "case refine_2.refine_2.refine_2\nR : Type u_1\nA : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nx y : ... | [
"case refine_2.refine_2.refine_2\nR : Type u_1\nA : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nx y : A ⊗[R] L\nz ... | mul_left_comm a₂ a₁ a₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.BaseChange | {
"line": 200,
"column": 8
} | {
"line": 200,
"column": 52
} | {
"line": 201,
"column": 8
} | [
{
"pp": "case refine_1\nR : Type u_1\nA : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nN : LieSubmodule R L M\n... | [
"case refine_1\nR : Type u_1\nA : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nN : LieSubmodule R L M\nx : A ⊗[R] L... | change toEnd A (A ⊗[R] L) (A ⊗[R] M) _ _ ∈ _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 495,
"column": 6
} | {
"line": 495,
"column": 15
} | {
"line": 495,
"column": 16
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : IsNilpotent L M\nh : Disjoint (lowerCentralSeries R L M 1) (maxTrivSubmodule R L M)\na✝ : Non... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : IsNilpotent L M\nh : Disjoint (lowerCentralSeries R L M 1) (maxTrivSubmodule R L M)\na✝ : Nontrivial M\nc... | h.eq_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finiteness | {
"line": 36,
"column": 4
} | {
"line": 41,
"column": 89
} | {
"line": 43,
"column": 0
} | [
{
"pp": "case mpr\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nb : Basis (↑(Basis.ofVectorSpaceIndex K V)) K V := ⋯\n⊢ (Basis.ofVectorSpaceIndex K V).Finite → IsNoetherian K V",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | intro hbfinite
refine
@isNoetherian_of_linearEquiv K K (⊤ : Submodule K V) V _ _ _ _ _ _ (RingHom.id K) _ _ _
(LinearEquiv.ofTop _ rfl) (id ?_)
refine isNoetherian_of_fg_of_noetherian _ ⟨Set.Finite.toFinset hbfinite, ?_⟩
rw [Set.Finite.coe_toFinset, ← b.span_eq, Basis.coe_ofVectorSpace, Subtyp... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Finiteness | {
"line": 36,
"column": 4
} | {
"line": 41,
"column": 89
} | {
"line": 43,
"column": 0
} | [
{
"pp": "case mpr\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nb : Basis (↑(Basis.ofVectorSpaceIndex K V)) K V := ⋯\n⊢ (Basis.ofVectorSpaceIndex K V).Finite → IsNoetherian K V",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | intro hbfinite
refine
@isNoetherian_of_linearEquiv K K (⊤ : Submodule K V) V _ _ _ _ _ _ (RingHom.id K) _ _ _
(LinearEquiv.ofTop _ rfl) (id ?_)
refine isNoetherian_of_fg_of_noetherian _ ⟨Set.Finite.toFinset hbfinite, ?_⟩
rw [Set.Finite.coe_toFinset, ← b.span_eq, Basis.coe_ofVectorSpace, Subtyp... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 591,
"column": 67
} | {
"line": 591,
"column": 88
} | {
"line": 593,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\n⊢ LieModule.lowerCentralSeries R L M k = ⊥ ↔ ucs k ⊥ = ⊤",
"ppTerm": "?m.68",
"assigned... | [] | simp [ucs_eq_top_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 591,
"column": 67
} | {
"line": 591,
"column": 88
} | {
"line": 593,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\n⊢ LieModule.lowerCentralSeries R L M k = ⊥ ↔ ucs k ⊥ = ⊤",
"ppTerm": "?m.68",
"assigned... | [] | simp [ucs_eq_top_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 591,
"column": 67
} | {
"line": 591,
"column": 88
} | {
"line": 593,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\n⊢ LieModule.lowerCentralSeries R L M k = ⊥ ↔ ucs k ⊥ = ⊤",
"ppTerm": "?m.68",
"assigned... | [] | simp [ucs_eq_top_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IntermediateField.Algebraic | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 45
} | {
"line": 129,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nF E : IntermediateField K L\nh : F ≤ E\nx✝ : Algebra ↥F ↥E := (inclusion h).toAlgebra\nthis : IsScalarTower (↥F) (↥E) L\n⊢ finrank (↥E) L ∣ finrank (↥F) L",
"ppTerm": "?m.60",
"assigned": true,
"usedConstan... | [] | exact Module.finrank_dvd_finrank_left F E L | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 25
} | {
"line": 149,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS T : IntermediateField F E\n⊢ Set.range ⇑(algebraMap F E) ∪ (↑S ∪ ↑T) = ↑S ∪ ↑T",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap",
"congrArg",
... | [
"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS T : IntermediateField F E\n⊢ Set.range ⇑(algebraMap F E) ⊆ ↑S ∪ ↑T"
] | rw [Set.union_eq_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 25
} | {
"line": 180,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set (IntermediateField F E)\nhS : S.Nonempty\nh : toSubfield '' S = Subfield.closure '' SetLike.coe '' S\n⊢ Set.range ⇑(algebraMap F E) ∪ ⋃₀ (SetLike.coe '' S) = ⋃₀ (SetLike.coe '' S)",
"ppTerm": "?m.109",
... | [
"F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set (IntermediateField F E)\nhS : S.Nonempty\nh : toSubfield '' S = Subfield.closure '' SetLike.coe '' S\n⊢ Set.range ⇑(algebraMap F E) ⊆ ⋃₀ (SetLike.coe '' S)"
] | rw [Set.union_eq_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 383,
"column": 28
} | {
"line": 383,
"column": 57
} | {
"line": 383,
"column": 58
} | [
{
"pp": "case a\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS T : Set E\n⊢ ↑(adjoin (↥(adjoin F S)) T) ⊆ ↑(adjoin F (S ∪ T))",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
... | [
"case a\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS T : Set E\n⊢ Set.range ⇑(algebraMap (↥(adjoin F S)) E) ⊆ ↑(adjoin F (S ∪ T)) ∧ T ⊆ ↑(adjoin F (S ∪ T))"
] | rw [adjoin_subset_adjoin_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 383,
"column": 28
} | {
"line": 383,
"column": 57
} | {
"line": 383,
"column": 58
} | [
{
"pp": "case a\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS T : Set E\n⊢ ↑(adjoin F (S ∪ T)) ⊆ ↑(adjoin (↥(adjoin F S)) T)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
... | [
"case a\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS T : Set E\n⊢ Set.range ⇑(algebraMap F E) ⊆ ↑(adjoin (↥(adjoin F S)) T) ∧ S ∪ T ⊆ ↑(adjoin (↥(adjoin F S)) T)"
] | rw [adjoin_subset_adjoin_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 715,
"column": 2
} | {
"line": 715,
"column": 54
} | {
"line": 717,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nf : L →ₐ[K] L'\nS : IntermediateField K L'\nh : S ≤ f.fieldRange\n⊢ map f (comap f S) = S",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [] | simpa only [inf_of_le_left h] using map_comap_eq f S | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 715,
"column": 2
} | {
"line": 715,
"column": 54
} | {
"line": 717,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nf : L →ₐ[K] L'\nS : IntermediateField K L'\nh : S ≤ f.fieldRange\n⊢ map f (comap f S) = S",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [] | simpa only [inf_of_le_left h] using map_comap_eq f S | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 715,
"column": 2
} | {
"line": 715,
"column": 54
} | {
"line": 717,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nf : L →ₐ[K] L'\nS : IntermediateField K L'\nh : S ≤ f.fieldRange\n⊢ map f (comap f S) = S",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [] | simpa only [inf_of_le_left h] using map_comap_eq f S | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 12
} | {
"line": 388,
"column": 4
} | [
{
"pp": "k : Type u\ninst✝¹⁸ : Field k\nK✝ : Type u\ninst✝¹⁷ : Field K✝\nL : Type v\nM : Type w\ninst✝¹⁶ : Field L\ninst✝¹⁵ : Algebra K✝ L\ninst✝¹⁴ : Field M\ninst✝¹³ : Algebra K✝ M\ninst✝¹² : IsAlgClosed M\ninst✝¹¹ : Algebra.IsAlgebraic K✝ L\nR : Type u\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\nS : Type v\ni... | [
"k : Type u\ninst✝¹⁸ : Field k\nK✝ : Type u\ninst✝¹⁷ : Field K✝\nL : Type v\nM : Type w\ninst✝¹⁶ : Field L\ninst✝¹⁵ : Algebra K✝ L\ninst✝¹⁴ : Field M\ninst✝¹³ : Algebra K✝ M\ninst✝¹² : IsAlgClosed M\ninst✝¹¹ : Algebra.IsAlgebraic K✝ L\nR : Type u\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\nS : Type v\ninst✝⁸ : Comm... | unfold f | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 456,
"column": 52
} | {
"line": 456,
"column": 90
} | {
"line": 456,
"column": 90
} | [
{
"pp": "k : Type u\ninst✝¹⁶ : Field k\nK : Type u_1\nJ : Type u_2\nR : Type u\nS : Type u_3\nL : Type v\nM : Type w\ninst✝¹⁵ : Field K\ninst✝¹⁴ : Field J\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Field L\ninst✝¹⁰ : Field M\ninst✝⁹ : Algebra R M\ninst✝⁸ : IsTorsionFree R M\ninst✝⁷ : IsAlgClosure R ... | [] | by simp [RingHom.algebraMap_toAlgebra] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 527,
"column": 6
} | {
"line": 527,
"column": 52
} | {
"line": 528,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝³ : Field F\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsAlgebraic F E\nn : ℕ\nL : IntermediateField F E\nfin : FiniteDimensional F ↥⊤\nhn : n < finrank F ↥L\nhnfd : ∀ (x : E), x ∈ L\n⊢ FiniteDimensional F E",
"ppTerm": "?m.104",
"assigned": true,
... | [] | exact topEquiv.toLinearEquiv.finiteDimensional | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Extension | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 68
} | {
"line": 270,
"column": 2
} | [
{
"pp": "case refine_1\nF : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁸ : Field F\ninst✝⁷ : Field E\ninst✝⁶ : Field K\ninst✝⁵ : Algebra F E\ninst✝⁴ : Algebra F K\nS : Set E\nL : Type u_4\ninst✝³ : Field L\ninst✝² : Algebra F L\ninst✝¹ : Algebra L E\ninst✝ : IsScalarTower F L E\nf : L →ₐ[F] K\nhK : ∀ s ∈ S, IsI... | [
"case refine_1\nF : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁸ : Field F\ninst✝⁷ : Field E\ninst✝⁶ : Field K\ninst✝⁵ : Algebra F E\ninst✝⁴ : Algebra F K\nS : Set E\nL : Type u_4\ninst✝³ : Field L\ninst✝² : Algebra F L\ninst✝¹ : Algebra L E\ninst✝ : IsScalarTower F L E\nf : L →ₐ[F] K\nhK : ∀ s ∈ S, IsIntegral L s ... | have : IsScalarTower L L' E := IsScalarTower.of_algebraMap_eq' rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 32
} | {
"line": 120,
"column": 4
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁴ : LieRing.IsNilpotent ↥H\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nχ₁ χ₂ χ₃ : ↥H → R\nhχ : χ₁ + χ₂ = χ₃\nt ... | [
"R : Type u_1\nL : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁴ : LieRing.IsNilpotent ↥H\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nχ₁ χ₂ χ₃ : ↥H → R\nhχ : χ₁ + χ₂ = χ₃\nt : R\nx : ↥(r... | simp only [RingHom.id_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 146,
"column": 2
} | {
"line": 149,
"column": 81
} | {
"line": 151,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁴ : LieRing.IsNilpotent ↥H\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nα χ : ↥H → R\nx : L\nhx : x ∈ rootSpace ... | [] | intro m hm
let x' : rootSpace H α := ⟨x, hx⟩
let m' : genWeightSpace M χ := ⟨m, hm⟩
exact (rootSpaceWeightSpaceProduct R L H M α χ (α + χ) rfl (x' ⊗ₜ m')).property | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 146,
"column": 2
} | {
"line": 149,
"column": 81
} | {
"line": 151,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁴ : LieRing.IsNilpotent ↥H\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nα χ : ↥H → R\nx : L\nhx : x ∈ rootSpace ... | [] | intro m hm
let x' : rootSpace H α := ⟨x, hx⟩
let m' : genWeightSpace M χ := ⟨m, hm⟩
exact (rootSpaceWeightSpaceProduct R L H M α χ (α + χ) rfl (x' ⊗ₜ m')).property | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.BilinearForm.Properties | {
"line": 168,
"column": 4
} | {
"line": 169,
"column": 52
} | {
"line": 170,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : BilinForm R M\nι : Type u_8\nb : Basis ι R M\nh : ∀ (i j : ι), (B (b i)) (b j) = (B (b j)) (b i)\nx y : M\nfx : M → R\ntx : Finset M\nix : ↑tx ⊆ Set.range ⇑b\nhx : ∑ a ∈ tx, fx a • a = x\n⊢ (B x) y = ... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : BilinForm R M\nι : Type u_8\nb : Basis ι R M\nh : ∀ (i j : ι), (B (b i)) (b j) = (B (b j)) (b i)\nx y : M\nfx : M → R\ntx : Finset M\nix : ↑tx ⊆ Set.range ⇑b\nhx : ∑ a ∈ tx, fx a • a = x\nfy : M → R\nty : Finset ... | obtain ⟨fy, ty, iy, -, hy⟩ := Submodule.mem_span_iff_exists_finset_subset.1
(by simp : y ∈ Submodule.span R (Set.range b)) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 80,
"column": 14
} | {
"line": 80,
"column": 16
} | {
"line": 81,
"column": 4
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nΦ : LinearMap.BilinForm R M\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nN : LieSubmodule R L M\nx : L\ny : M\nH : ... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nΦ : LinearMap.BilinForm R M\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nN : LieSubmodule R L M\nx : L\ny : M\nH : ∀ n ∈ N, (Φ ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.BooleanGenerators | {
"line": 91,
"column": 2
} | {
"line": 103,
"column": 23
} | {
"line": 105,
"column": 0
} | [
{
"pp": "case h.right\nα : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nC : Set α\nhC : ∀ x ∈ C, IsCompactElement x\nha : sSup C ≤ sSup S\nT : (b : α) → IsCompactElement b → b ≤ sSup S → Set α\nhT₁ : ∀ (b : α) (a : IsCompactElement b) (a_1 : b ≤ sSup... | [] | · apply le_antisymm
· apply _root_.sSup_le
intro c hc
rw [hT₂ c (hC _ hc) ((le_sSup hc).trans ha)]
apply sSup_le_sSup
apply _root_.le_sSup
use c, hc, hC _ hc, (le_sSup hc).trans ha
· simp only [Set.sSup_eq_sUnion, sSup_le_iff, Set.mem_sUnion, Set.mem_ofPred_eq,
forall_exist... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.BooleanGenerators | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 14
} | {
"line": 180,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nX Y : Set α\nhX : X ⊆ S\nhY : Y ⊆ S\nh : sSup X ≤ sSup Y\na : α\n⊢ a ∈ X → a ∈ Y",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.instMe... | [
"α : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nX Y : Set α\nhX : X ⊆ S\nhY : Y ⊆ S\nh : sSup X ≤ sSup Y\na : α\nha : a ∈ X\n⊢ a ∈ Y"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 325,
"column": 2
} | {
"line": 326,
"column": 38
} | {
"line": 327,
"column": 2
} | [
{
"pp": "V : Type u_5\nK : Type u_6\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb₁ : B.IsRefl\nb₂ : (B.restrict W).Nondegenerate\nthis : W ⊓ B.orthogonal W = ⊥\n⊢ finrank K V ≤ finrank K ↥(W ⊔ B.orthogonal W) + 0",
"pp... | [
"V : Type u_5\nK : Type u_6\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb₁ : B.IsRefl\nb₂ : (B.restrict W).Nondegenerate\nthis : W ⊓ B.orthogonal W = ⊥\n⊢ finrank K V ≤ finrank K V + finrank K ↥(W ⊓ B.orthogonal ⊤)"
] | rw [← finrank_bot K V, ← this, finrank_sup_add_finrank_inf_eq,
finrank_add_finrank_orthogonal b₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.BooleanGenerators | {
"line": 187,
"column": 10
} | {
"line": 187,
"column": 12
} | {
"line": 188,
"column": 2
} | [
{
"pp": "case a\nα : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nh : sSup S = ⊤\na : α\n⊢ a ∈ {a | IsAtom a} → a ∈ S",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Set.ofPred",
"PartialOrder.toPreorder",
"Preo... | [
"case a\nα : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nh : sSup S = ⊤\na : α\nha : a ∈ {a | IsAtom a}\n⊢ a ∈ S"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Data.Multiset.Fintype | {
"line": 123,
"column": 8
} | {
"line": 123,
"column": 10
} | {
"line": 123,
"column": 10
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\ns : Finset (α × ℕ)\nhsm : s ⊆ m.toEnumFinset\na : α\nha : (filter (fun x ↦ a = x.1) s.val).card = 0\n⊢ (filter (fun a_1 ↦ a = a_1.1) s.val).card ≤ count a m",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"case inl\nα : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\ns : Finset (α × ℕ)\nhsm : s ⊆ m.toEnumFinset\na : α\nha : (filter (fun x ↦ a = x.1) s.val).card = 0\n⊢ 0 ≤ count a m"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Killing | {
"line": 145,
"column": 45
} | {
"line": 145,
"column": 72
} | {
"line": 146,
"column": 6
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁴ : Field K\ninst✝³ : LieRing L\ninst✝² : LieAlgebra K L\ninst✝¹ : IsKilling K L\ninst✝ : Module.Finite K L\nI : LieIdeal K L\nthis : Disjoint I (killingCompl K L I)\n⊢ IsCompl ↑I ↑(killingCompl K L I)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": ... | [
"K : Type u_2\nL : Type u_3\ninst✝⁴ : Field K\ninst✝³ : LieRing L\ninst✝² : LieAlgebra K L\ninst✝¹ : IsKilling K L\ninst✝ : Module.Finite K L\nI : LieIdeal K L\nthis : Disjoint I (killingCompl K L I)\n⊢ IsCompl (↑I) ((killingForm K L).orthogonal ↑I)"
] | I.toSubmodule_killingCompl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 107,
"column": 31
} | {
"line": 125,
"column": 56
} | {
"line": 127,
"column": 0
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁵ : LieRing L\ninst✝⁴ : Field K\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : ↥H → K\nx y : L\nhx : x ∈ rootSpace H α\nhy : y ∈ rootSpace H β\nhαβ : α + β ≠ 0\n⊢... | [] | by
/- If `ad R L z` is semisimple for all `z ∈ H` then writing `⟪x, y⟫ = killingForm K L x y`, there
is a slick proof of this lemma that requires only invariance of the Killing form as follows.
For any `z ∈ H`, we have:
`α z • ⟪x, y⟫ = ⟪α z • x, y⟫ = ⟪⁅z, x⁆, y⟫ = - ⟪x, ⁅z, y⁆⟫ = - ⟪x, β z • y⟫ = - β z • ⟪x, y⟫... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 385,
"column": 6
} | {
"line": 386,
"column": 68
} | {
"line": 387,
"column": 4
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : PerfectField K\nα : ↥H → K\nx : L\nh : ↥H\nhx : x ∈ ((ad... | [
"K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : PerfectField K\nα : ↥H → K\nx : L\nh : ↥H\nhx : x ∈ ((ad K L) ↑h).ei... | (isSemisimple_ad_of_mem_isCartanSubalgebra
h.property).isFinitelySemisimple.maxGenEigenspace_eq_eigenspace, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 570,
"column": 2
} | {
"line": 573,
"column": 34
} | {
"line": 574,
"column": 2
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\ne : L... | [
"K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\ne : L\nheα : e ∈ ... | replace hef : ⁅⁅e, f⁆, e⁆ = 2 • e := by
have : ⁅⁅e, f'⁆, e⁆ = α h • e := lie_eq_smul_of_mem_rootSpace heα h
rw [lie_smul, smul_lie, this, ← smul_assoc, smul_eq_mul, mul_assoc, inv_mul_cancel₀ hh,
mul_one, two_smul, two_smul] | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 598,
"column": 2
} | {
"line": 600,
"column": 36
} | {
"line": 601,
"column": 2
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\ne f :... | [
"K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\ne f : L\nheα : e ... | have h_eq : h = killingForm K L e f • α' := by
simp only [hα', Subtype.ext_iff, ← ht.lie_e_f, hef]
rw [Submodule.coe_smul_of_tower] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 591,
"column": 51
} | {
"line": 596,
"column": 73
} | {
"line": 598,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ eval 2 (C R n) = 2",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.instAddMonoidWithOne",
"HMul.hMul",
"Mathl... | [] | by
induction n using Polynomial.Chebyshev.induct with
| zero => simp
| one => simp
| add_two n ih1 ih2 => simp [C_add_two, ih1, ih2]; norm_num
| neg_add_one n ih1 ih2 => simp [C_sub_one, -C_neg, ih1, ih2]; norm_num | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 277,
"column": 2
} | {
"line": 278,
"column": 50
} | {
"line": 280,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j : ι\n⊢ (P.reflectionPerm i) ((P.reflectionPerm i) j) = j",
"ppTerm": "?m.31",
"assigned": true,
... | [] | apply P.root.injective
simp only [root_reflectionPerm, reflection_same] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 277,
"column": 2
} | {
"line": 278,
"column": 50
} | {
"line": 280,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\ni j : ι\n⊢ (P.reflectionPerm i) ((P.reflectionPerm i) j) = j",
"ppTerm": "?m.31",
"assigned": true,
... | [] | apply P.root.injective
simp only [root_reflectionPerm, reflection_same] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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